Primal-dual algorithms for next-generation radio-interferometric imaging

Transcription

1 1/17 Primal-dual algorithms for next-generation radio-interferometric imaging Alexandru Onose 1,RafaelE.Carrillo 2,AbdullahAbdulaziz 1, Arwa Dabbech 1,JasonD.McEwen 3 and Yves Wiaux 1 1 Institute of Sensors, Signals and Systems, Heriot-Watt University, United Kingdom 2 Signal Processing Laboratory, Ecole Polytechnique Fédérale de Lausanne, Switzerland 3 Mullard Space Science Laboratory, University College London, United Kingdom

2 Next-generation radio-interferometric challenges 2/17 Increase the resolution and sensitivity up to two orders of magnitude over current instruments gigapixel images huge dynamic range

3 Next-generation radio-interferometric challenges 2/17 Increase the resolution and sensitivity up to two orders of magnitude over current instruments gigapixel images huge dynamic range Unprecedented amount of data to be processed Good reconstruction quality with scalable algorithms employing parallel and distributed processing

4 Measurement model Measurement equation yu) = Dl, u)xl)e 2iπu l d 2 l 3/17 Discretised version of the inverse problem y = Φx + n with Φ = GFZ x R N + the intensity image of interest Φ C M N alinearmap;imagedomaintovisibilityspace y C M the measured visibilities G C M kn gridding matrix modelling DDEs FZ C kn N Fourier matrix with zero padding

5 Measurement model Measurement equation yu) = Dl, u)xl)e 2iπu l d 2 l 3/17 Discretised version of the inverse problem y = Φx + n with Φ = GFZ x R N + the intensity image of interest Φ C M N alinearmap;imagedomaintovisibilityspace y C M the measured visibilities G C M kn gridding matrix modelling DDEs FZ C kn N Fourier matrix with zero padding extremely large M and N

6 Measurement model Measurement equation yu) = Dl, u)xl)e 2iπu l d 2 l 3/17 Discretised version of the inverse problem y = Φx + n with Φ = GFZ x R N + the intensity image of interest Φ C M N alinearmap;imagedomaintovisibilityspace y C M the measured visibilities G C M kn gridding matrix modelling DDEs FZ C kn N Fourier matrix with zero padding extremely large M and N scalable algorithms; parallelisms and distributed processing

7 Convex optimisation and compressive sensing Problem formulation 0) 4/17 Discretised version of the ill-posed inverse problem y = Φx + n with Φ = GFZ Compressive sensing approach Constrain the solution to belong to the intersection of several convex sets enforcing data fidelity and positivity Add a convex regulariser on the solution to enforce low dimensionality, sparsity,inagiventransform Convex optimisation solvers Primal-dual forward-backward algorithm Distributed single-band RI imaging Wide-band RI imaging

8 Distributed single-band RI imaging Problem formulation 1) Split the large-scale inverse problem block wise 5/17 y = Φ x + n with Φ = G FZ y = y 1. Φ = Φ 1. = G 1 M 1. FZ y nd Φ nd G nd M nd Regularisation of the ill-posed problem Sparsity constraint for x in a collection of wavelet bases ] Ψ = [Ψ 1... Ψ nb

9 Distributed single-band RI imaging Problem formulation 2) 6/17 Convex optimisation task min x f x) + γ n b i=1 l i Ψ i x) + n d =1 h Φ x) Enforce positivity, sparsity and data fidelity f z) =ι C z), C = R N + l i z) = z 1 h z) =ι B z), B = {z C M : z y 2 ϵ }

10 Distributed single-band RI imaging The primal dual approach Primal problem 7/17 min f x)+γ x n b i=1 l i Ψ n d i x)+ h Φ x) =1 Dual formulation of the original convex optimisation task min f u i v n b i=1 Ψ i u i Primal dual algorithm n d =1 Φ v ) + 1 γ n b i=1 l i u i)+ n d =1 h v ) Alternate solving the primal problem and the dual problem Converges towards a Kuhn-Tucker point

11 Distributed single-band RI imaging Advantages of the primal dual approach 8/17 Full splitting of the operators and functions No inversion of the linear operators The updates are performed on the dual variables in parallel Non smooth convex functions Forward backward iterations Alternate between a gradient like step forward) and a proximal update backward) Randomised updates on the dual variables Reduce computational and memory requirements per iteration Require more iterations to converge

12 Distributed single-band RI imaging Primal dual algorithm given x 0), x 0), u 0), v 0), ũ 0), ṽ 0),γ,τ,σ i i i repeat for t =1,... generate sets P {1,...,n b } and D {1,...,n d } 9/17 b t) = M FZ x t 1), D run simultaneously Ddistribute b t) and do in parallel v t) = I P B ) v t 1) + G b t) ) ṽ t) = G vt) i end and gather ṽ t) i Pdo in parallel end end u t) i = I S γ σ i x t) = P C x t 1) τ x t) =2 x t) x t 1) until convergence ) u t 1) n b i=1 i + Ψ i x t)) ũ t) i σ i ũ t) + Z F i n d =1 = Ψ i u t) i ) ς M ṽt)

13 Distributed single-band RI imaging Computational complexity 10 / 17 b t) = M FZ x t 1), D Ddistribute b t) and do in parallel v t) = I P B ) v t 1) + G b t) ) ṽ t) = G vt) end and gather ṽ t) i Pdo in parallel end u t) i = I S γ σ i ) u t 1) x t) = P C... i ) + Ψ i x t)) ũ t) i = Ψ i u t) i central node O p Pi O ) O kn log kn + O kn log kn 2n b N ) ) n b + k)n + n d n v ) n d data fidelity nodes p D O 2 kns n d MN + M )

14 Distributed single-band RI imaging Some results 11 / SNR number of iterations PD,M = 10N ADMM,M = 10N SDMM,M = 10N PD,M =5N ADMM,M =5N SDMM,M =5N PD,M =2N ADMM,M =2N SDMM,M =2N SNR number of iterations PD PD R,pP =1,pD =0.9 PD R,pP =1,pD =0.8 PD R,pP =1,pD =0.7 PD R,pP =1,pD =0.6 PD R,pP =1,pD =0.5 PD R,pP =1,pD =0.4 PD R,pP =1,pD =0.3 PD R,pP =1,pD = R. E. Carrillo, J. D. McEwen, and Y. Wiaux, PURIFY: a new approach to radio-interferometric imaging, MNRAS, vol. 439, no. 4, pp , 2014.

15 Distributed single-band RI imaging Some results 12 / SNR Complexity normalised) number of iterations PD PD R,pP =1,pD =0.9 PD R,pP =1,pD =0.8 PD R,pP =1,pD =0.7 PD R,pP =1,pD =0.6 PD R,pP =1,pD =0.5 PD R,pP =1,pD =0.4 PD R,pP =1,pD =0.3 PD R,pP =1,pD =0.2 2 PD R PD R only FFT) 1.5 PD R only data term) pd

16 Wide-band RI imaging Problem formulation 0) 13 / 17 Multiple frequency bands; images x l at each band l =1,...,L Discretised version of the ill-posed inverse problem y l = Φ l x l + n l with Φ = G l FZ X =[x 1,...,x L ] R N L + the hyper-spectral image cube Y =[y 1,...,y L ] C M L the measured visibilities Φ l C M N alinearmap;imagedomaintovisibilityspace Compressive sensing approach Enforce data fidelity and positivity Convex regulariser to enforce low dimensionality Joint sparsity and low rank on X Primal-dual forward-backward algorithm

17 Wide-band RI imaging Problem formulation 1) 14 / 17 Minimisation problem min X f X) + µ g 1Ψ X) + g 2 X) + L l=1 h l Φl x l ) ) Enforce positivity, oint sparsity, low rank, data fidelity f Z) =ι D Z), D = R N L + g 1 Z) = Z l2,1, g 2 Z) = Z h I z) =ι Bl z), B l = {z C M : z y l 2 ϵ l }

18 Wide-band RI imaging Primal dual algorithm given X 0), X 0), V 0) 1, V0) 2, V0) 3,µ,τ,σ 1,σ 2,σ 3 repeat for t =1,... do in parallel end end V t) t 1) 1 =V 1 + Ψ X t 1) S l 2,1 V t 1) µ/σ Ψ X t 1)) V t) 2 =Vt 1) 2 + X t 1) S 1/σ 2 V t 1) 2 + X t 1)) l {1,...,L} do in parallel u t) l X t) =P C X t 1) τ =u t 1) + Φ l l x t 1) P l Bl u t 1) l ) + Φ l x t 1) l [ ])) σ 1 ΨV t) 1 + σ 2 V t) 2 + σ 3 Φ 1 ut) 1,...,Φ L ut) L 15 / 17 X t) =2X t) X t 1) until convergence

19 Distributed single-band RI imaging Some results 16 / SNR LRJS, 64-bands LRJS, 32-bands LRJS, 16-bands WDCT, 64-bands WDCT, 32-bands WDCT, 16-bands Single-band average, 64-bands Single-band average, 32-bands Single-band average, 16-bands M/N -3 A. Ferrari, J. Deguignet, C. Ferrari, D. Mary, A. Schutz, and O. Smirnov, Multi-frequency image reconstruction for radio interferometry. A regularized inverse problem approach, ArXiv e-prints, Apr

20 Summary 17 / 17 Full splitting of the operators and functions No inversion of the linear operators Broad range of convex prior functions Forward backward iterations for non smooth functions Randomisation Reduce computational and memory requirements Require more iterations to converge